Suppose A is a nice abelian category (such as coherent sheaves coh(X) on a smooth complex projective variety X, or representations mod-CQ of a quiver Q) or T is a nice triangulated category (such as D^bcoh(X) or D^bmod-CQ) over C. Let M be the moduli stack of objects in A or T. Consider the homology H_*(M) over some ring R.
  Given a little extra data on M, for which there are natural choices in our examples, I will explain how to define the structure of a graded vertex algebra on H_*(M). By a standard construction, one can then define a graded Lie algebra from the vertex algebra; roughly speaking, this is a Lie algebra structure on the homology H_*(M^{pl}) of a "projective linear” version M^{pl} of the moduli stack M.
  For example, if we take T = D^bmod-CQ, the vertex algebra H_*(M) is the lattice vertex algebra attached to the dimension vector lattice Z^{Q_0} of Q with the symmetrized intersection form. The degree zero part of the graded Lie algebra contains the associated Kac-Moody algebra.
  The construction appears to be new, but is connected with a lot of work in Geometric Representation Theory, to do with Ringel-Hall-type algebras and their representations, such as the results of Grojnowski-Nakajima on Hilbert schemes. The vertex algebra construction is enormously general, and applies in huge classes of examples. There is a differential-geometric version too.
  The question I am hoping someone in the audience will answer is this: what is the physical interpretation of these vertex algebras?
  It is in some sense an "even Calabi-Yau” construction: when applied to coh(X) or D^bcoh(X), it is most natural for X a Calabi-Yau 2-fold or Calabi-Yau 4-fold, and is essentially trivial for X a Calabi-Yau 3-fold. I discovered it when I was investigating wall-crossing for Donaldson-Thomas type invariants for Calabi-Yau 4-folds. So perhaps one should look for an explanation in the physics of Calabi-Yau 2-folds or 4-folds, with M the moduli space of boundary conditions for the associated SCFT.

I will review the concept of duality in quantum systems from the 2D Ising model to superconformal field theories in higher dimensions. Using some of these latter theories, I will explain how a generalized concept of duality emerges: these are dualities not between full theories but between algebraically well-defined sub-sectors of strikingly different theories.

Given a smooth variety X containing a smooth divisor Y, the relative Gromov-Witten invariants of (X,Y) are defined as certain counts of algebraic curves in X with specified orders of tangency to Y. Their intrinsic interest aside, they are an important part of any Gromov-Witten theorist’s toolkit, thanks to their role in the celebrated “degeneration formula.” In recent years these invariants have been significantly generalised, using techniques in logarithmic geometry. The resulting “log Gromov-Witten invariants” are defined for a large class of targets, and in particular give a rigorous definition of relative invariants for (X,D) where D is a normal crossings divisor. Besides being more general, these numbers are  intimately related to constructions in Mirror Symmetry, via the Gross-Siebert program. In this talk, we will describe a recursive formula for computing the invariants of (X,D) in genus zero. The result relies on a comparison theorem which expresses the log Gromov-Witten invariants as classical (i.e. non log-geometric) objects.
 

In this talk, first we  address the convergence issues of a standard finite volume element method (FVEM) applied to simple elliptic problems. Then, we discuss discontinuous finite volume element methods (DFVEM) for elliptic problems  with emphasis on  computational and theoretical  advantages over the standard FVEM. Further, we present a natural extension of DFVEM employed for the elliptic problem to the Stokes problems. We also discuss suitability of these methods for the approximation of incompressible miscible displacement problems.
 

Both in the real and in the p-adic case, I will talk about recent results about C^r-parameterizations and their diophantine applications.  In both cases, the dependence on r of the number of parameterizing C^r maps plays a role. In the non-archimedean case, we get as an application new bounds for rational points of bounded height lying on algebraic varieties defined over finite fields, sharpening the bounds by Sedunova, and making them uniform in the finite field. In the real case, some results from joint work with Pila and Wilkie, and also beyond this work, will be presented, 
in relation to several questions raised by Yomdin. The non-archimedean case is joint work with Forey and Loeser. The real case is joint work with Pila and Wilkie, continued by my PhD student S. Van Hille.  Some work with Binyamini and Novikov in the non-archimedean context will also be mentioned. The relations with questions by Yomdin is joint work with Friedland and Yomdin. 

This lecture will give a brief review of the theory of non-abelian reciprocity maps and their applications to Diophantine geometry, and pose some questions for model-theorists.
 

Several works (by Kontsevich, Soibelman, Berkovich, Nicaise, Boucksom, Jonsson...) have shown that the limit behavior of a one-parameter family $(X_t)$ of complex algebraic varieties can often be described using the associated Berkovich t-adic analytic space $X^b$. In a work in progress with E. Hrushovski and F. Loeser, we provide a new instance of this general phenomenon. Suppose we are given for every t an  $(n,n)$-form $ω_t$ on $X_t$ (for n= dim X). Then under some assumptions on the formula that describes $ω_t$, the family $(ω_t)$ has a "limit" ω, which is a real valued  (n,n)-form in the sense of Chambert-Loir and myself on the Berkovich space $X^b$, and the integral of $ω_t$ on $X_t$ tends to the integral of ω on $X^b$. 
In this talk I will first make some reminders about Berkovich spaces and (n,n)-forms in this setting, and then discuss the above result. 
In fact, as I will explain, it is more convenient to formulate it with  $(X_t)$ seen as a single algebraic variety over a non-standard model *C of C and (ω_t) as a (n,n) differential form on this variety. The field *C also carries a t-adic real valuation which makes it a model of ACVF (and enables to do Berkovich geometry on it), and our proof uses repeatedly RCF and ACVF theories. 
 

I will report on two recent papers with D. Ghioca and U. Zannier (joined by P. Corvaja and F. Hu, respectively) in which we consider variants of the Mordell-Lang conjecture.  In the first of these, we study the dynamical Mordell-Lang conjecture in positive characteristic, proving some instances, but also showing that in general the problem is at least as hard as a difficult diophantine problem over the integers.  In the second paper, we study the Mordell-Lang problem for extensions of abelian varieties by the additive group.  Here we have positive results in the function field case obtained by using the socle theorem in the form offered as an aside in Hrushovski's 1996 paper and in the number field case we relate this problem to the Bombieri-Lang conjecture.